The partially ordered set of one-point extensions

A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y’$ of $X$ let $Y\leq Y’$ if there is a continuous function of $Y’$ into $Y$ which fixes $X$ point-wise. An extension $Y$ of $X$ is called a {\em one-point extension} of $X$ if $Y\backslash X$ is a singleton. Let ${\mathcal P}$ be a topological property. An extension $Y$ of $X$ is called a {\em ${\mathcal P}$-extension} of $X$ if it has ${\mathcal P}$. One-point ${\mathcal P}$-extensions comprise the subject matter of this article. Here ${\mathcal P}$ is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space $X$ (partially ordered by $\leq$) and the set of compact non-empty subsets of its outgrowth $\beta X\backslash X$ (partially ordered by $\subseteq$). This enables us to study the order-structure of various sets of one-point extensions of the space $X$ by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces $X$ denote by ${\mathscr U}(X)$ the set of all zero-sets of $\beta X$ which miss $X$. \noindent{\bf Conjecture.} {\em For locally compact spaces $X$ and $Y$ the partially ordered sets $({\mathscr U}(X),\subseteq)$ and $({\mathscr U}(Y),\subseteq)$ are order-isomorphic if and only if the spaces ${\em cl}{\beta X}(\beta X\backslash\upsilon X)$ and ${\em cl}{\beta Y}(\beta Y\backslash\upsilon Y)$ are homeomorphic.}

💡 Research Summary

The paper investigates the partially ordered set of one‑point extensions of a topological space X. An “extension” Y of X is a space containing X as a dense subspace; two extensions are considered equivalent if there exists a homeomorphism fixing X pointwise. On the set of equivalence classes a relation Y ≤ Y′ is defined: Y ≤ Y′ iff there is a continuous map from Y′ onto Y that is the identity on X. This relation is not the usual inclusion but rather a comparison of how much “extra” information each extension adds.

The authors focus on one‑point extensions, i.e., extensions for which Y \ X consists of a single point. For a Tychonoff space X they prove a striking anti‑order‑isomorphism between the collection 𝔈₁(X) of one‑point Tychonoff extensions (ordered by ≤) and the family 𝔎(βX \ X) of non‑empty compact subsets of the outgrowth βX \ X (ordered by inclusion). The map φ sends a one‑point extension Y, identified with X∪{p} for a point p∈βX \ X, to the compact set {p}. The anti‑isomorphism property means that Y ≤ Y′ exactly when φ(Y) ⊇ φ(Y′). Consequently, the order structure of one‑point extensions is completely captured by the inclusion order of compact subsets of the remainder of the Stone–Čech compactification.

The paper then introduces the notion of a 𝒫‑extension: an extension possessing a prescribed topological property 𝒫 (e.g., realcompactness, completeness, metacompactness). Under mild assumptions on 𝒫 (essentially that 𝒫 is hereditary for closed subspaces), the anti‑order‑isomorphism restricts to the subposet of one‑point 𝒫‑extensions. Thus, studying the order of such extensions reduces to studying the inclusion order of compact subsets of βX \ X that themselves satisfy the corresponding 𝒫‑condition in the ambient space βX.

A central object in the later part of the paper is the family
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